FACOLT `A DI SCIENZE MATEMATICHE FISICHE E NATURALI Corso di Laurea Magistrale in Fisica
Master Degree Thesis
Theoretical description
of few-nucleon scattering states
in terms of integral relations
Candidate:
Ylenia Capitani
Supervisor:
Prof. Laura Elisa Marcucci
1 Introduction 1
1.1 Potential models . . . 3
2 The HH method for the A-body system 7 2.1 Jacobi and hyperspherical coordinates . . . 7
2.2 Hyperspherical harmonic functions . . . 10
2.3 Bound state problem . . . 13
2.3.1 Bound state wave function . . . 13
2.3.2 Rayleigh-Ritz variational principle . . . 18
2.4 Scattering state problem . . . 21
2.4.1 Scattering state wave function . . . 21
2.4.2 Kohn variational principle . . . 24
2.4.3 Kohn variational principle in terms of integral relations . . . 26
2.4.4 Another derivation of the integral relations from the Kohn variational principle . . . 27
3 Two-Nucleon system 31 3.1 Jacobi coordinates . . . 31
3.2 Bound state problem . . . 32
3.3 Scattering state problem . . . 36
3.3.1 Integral relations . . . 37
Argonne v18 potential . . . 37
N3LO Idaho potential . . . 47
4 Three-Nucleon system 53 4.1 Jacobi and hyperspherical coordinates . . . 53
4.2 Bound state problem . . . 56
4.3 Scattering state problem . . . 59
4.3.1 Integral relations . . . 61
Argonne v14 potential . . . 61
5 Conclusions and outlook 73 A Useful functions 77 A.1 Jacobi polynomials . . . 77
A.2 Generalized Laguerre polynomials . . . 77
A.3 Spherical Bessel functions . . . 78
B Variational principles 79
B.1 Rayleigh-Ritz variational principle . . . 79
B.2 Kohn variational principle . . . 79
C Integral relations: Gaussian potential 81
D The secant method 85
E Phase shifts and mixing angles 87
F Details of calculation 89
3.1 Lowest eigenvalues for the states Jπ = 0+ (left) and Jπ = 0− (right) as a function of the parameter β, calculated using the AV18 potential. The dashed horizontal lines correspond to two fixed energies, E¯n =
0.5, 2.5 MeV, at which the phase shifts and the mixing angles are calculated. . . 38
3.2 Lowest eigenvalues for the states Jπ = 1+ (left) and Jπ = 1− (right) calculated with the AV18 potential. The lowest eigenvalue of the state Jπ = 1+ (left) is negative and it corresponds to the bound
np state of the deuteron: Ed = −2.224575 (MeV) [7]. The levels
organize in pairs (red/blue solid lines on the left, red-solid/yellow-dot lines on the right). The separation between levels is very small for the state Jπ = 1− and cannot be appreciated in the figure. The dashed horizontal lines correspond to two fixed energies, En¯ = 0.5, 2.5 MeV, at which phase shifts and mixing angles are calculated. With the help of these dashed horizontal lines, two sequential values of
β, belonging to lines with different colours (i.e. different asymptotic
scattering configurations), can be selected. . . 39
3.3 The same as Fig. 3.2 for the states Jπ = 2+ (left) and Jπ = 2− (right), still obtained with the AV18 potential. The levels organize in pairs (blue-solid/light-blue-dot lines on the left, red/blue solid lines on the right). The separation between levels is very small for the states Jπ = 2+. With the help of the dashed horizontal lines, two sequential values of β belonging to lines with different colours (i.e. different asymptotic scattering configurations) can be selected. . . . 39
3.4 Phase shifts 3S1, 3D1 and mixing parameter ε1 obtained with the AV18 potential at En¯ = 5 MeV as a function of the regularization parameter γ. The red lines are the corresponding results of Ref. [7]. The left and right panels correspond to the first- and second-order, respectively. . . 46
3.5 Same as the left panel of Fig.3.2but for the N3LO potential. . . 47
3.6 Same as Fig.3.4but for the N3LO potential [9]. . . 51
4.1 A = 3 internal Jacobi coordinates (x(p)2 , x(p)1 ) relative to the three possible permutations (p) of the three particles. From left, (¯p) = (3)
(reference permutation), (p) = (1) and (p) = (2). . . . 55
4.2 Lowest eigenvalues for the state Jπ = 12+ as a function of the pa-rameter β with Nα = 3, NL = 18 and for different values of Kαmax.
The AV14 potential is used. The dashed horizontal lines correspond to Enlab¯ = 2, 3 MeV (lower and upper, respectively, see the text for the details), at which the phase shifts and the mixing angles are calculated. The dot line represents the energy of the deuteron
Ed = −2.2263 MeV, while the solid black line is the energy of the
bound state of the system. . . 64
4.3 Same as Fig. 4.2 but with Nα = 8. The dashed horizontal lines
correspond to E¯nlab= 2, 3 MeV (lower and upper, respectively, see the text for the details), at which the phase shifts and the mixing angles are calculated. With the help of these lines two sequential values of β belonging to different asymptotic states (i.e. red/blue lines for
Kαmax = 160) can be selected. The dot horizontal line indicates the energy of the deuteron Ed= −2.2263 MeV, while the solid black line corresponds to the energy of the bound state of the system. . . 66
4.4 Lowest eigenvalues for the state Jπ = 12+as a function of the param-eter β with Nα = 18, NL = 20 and the following configuration for
the grand-angular momentum: Kαmax = 160 for the first eight chan-nels, Kαmax = 20 for the others. The AV14 potential model is used. The solid red (blue) lines correspond to S-wave (D-wave) asymptotic states. The horizontal dashed lines correspond to E¯nlab = 2, 3 MeV (lower and upper, respectively, see the text for the details), at which the phase shifts and the mixing angles are calculated. With the help of these lines two sequential values of β belonging to different asymp-totic states (i.e. red/blue lines) can be selected. The dot horizontal line represents the energy of the deuteron Ed= −2.2263 MeV, while
the solid black line corresponds to the energy of the bound state of the system. . . 67
4.5 Phase shifts2S1 2
,4D1 2
and mixing parameter η1 2
+ obtained with the AV14 potential at energy E¯nlab= 2 MeV as a function of the regular-ization parameter γ. The left and right panels correspond to the first-and second-order, respectively. The red first-and blue lines are the cor-responding results obtained in Ref. [2] and obtained using the KVP at second order, respectively. Apart from the 2S1
2 phase shift, they essentially coincide. . . 71
4.6 Same as Fig.4.5but at Enlab¯ = 3 MeV. . . 71
C.1 Phase shift tan δ1, calculated with the integral relations, as a function of the regularization parameter γ, for different values of NL. . . 84
3.1 N N states in spectroscopic notation with their relative quantum
numbers and parity π = (−1)L for total angular momentum J ≤ 3.
L + S + T must be odd for np states while for nn (pp) states L + S
is even (T = 1). 3S1−3D1, 3P2−3F2 and 3D3−3G3 are “coupled
states”. . . 34
3.2 S-wave and D-wave (PS and PD), or P -wave and F -wave (PP and PF), or singlet and triplet (PS=0and PS=1) occupation probabilities (%) for the first six energy eigenvalues En (energies are in MeV) calculated at fixed β = 3.0 fm−1, for different np Jπ states, using the AV18 potential. . . 40
3.3 T = 1 phase shifts and mixing parameters (in degrees) for np scat-tering using the AV18 potential, at fixed energies En¯ (in MeV). The energy En¯ available in the center-of-mass system corresponds to the energy in the laboratory system Enlab¯ = 2En¯. For every fixed en-ergy En¯, the scattering parameters in the first and second lines are the first- and second-order estimates, respectively. See text for more details. . . 43
3.4 Same as Tab.3.3but for the T = 0 phase shifts and mixing parameters. 44 3.5 Phase shifts and mixing parameters (in degrees) for the AV18 poten-tial at three different values of the energy En¯ = 5, 50, 100 MeV as a function of the dimension of the Laguerre basis (NL). At a given energy, the first and the second lines correspond to the first- and second-order estimates, respectively. The results from the calcula-tion from the KVP at the second order (KVPII) and from Ref. [7] are also listed. . . 45
3.6 Same as Tab.3.2 but for the N3LO potential. . . 48
3.7 Same as Tab.3.3 but for the N3LO potential. . . 49
3.8 Same as Tab.3.4 but for the N3LO potential. . . 50
3.9 Same as Tab.3.5 but for the N3LO potential [9]. . . 51
4.1 Expansion channels for the three-nucleon system wave function for the Jπ = 12+ state. The notation for the quantum numbers is given in the text. . . 59
4.2 Scattering channels for a general scattering process N Y → N Y with sN = 12, SY = 1 and fixed J = 12,32. The parity of the state is π = (−1)L. . . 60
4.3 S-, P - and D-wave occupation probabilities (PS, PP and PD) in %
for the negative eigenvalues En (energies are in MeV) calculated at fixed β = 2.4 fm−1, for the state Jπ = 12+, with Nα = 3, NL = 18
and Kαmax= 160. The AV14 potential is used. . . 64
4.4 Same as Tab.4.3 but with Nα= 8. . . 66
4.5 Same as Tab.4.3 but with Nα= 18 and NL= 20. . . 67
4.6 Scattering parameters (in degrees) calculated at first- and second-order at the energy Enlab¯ = 2 MeV with NL = 18, 20 and with increasing number of channels Nα up to a maximum of Nα = 18.
For Nα= 12, 14, 18 two different configurations of maximum
grand-angular momentum Kαmax for each group of channels Gi, i = 1, . . . , 5 have been taken in account (see text for more details). The AV14 potential is used. . . 69
4.7 Same as Tab.4.6 but at energy Enlab¯ = 3 MeV with NL= 20. . . 69
4.8 Scattering parameters (in degrees) calculated at first- and second-order at the energies Enlab¯ = 2, 3 MeV as a function of the Laguerre basis dimension NL, using Nα= 18. The AV14 potential is used. . . 70
4.9 Comparison between the Enlab¯ = 2 MeV scattering parameters at first-and second-order obtained with the integral relations (with regular-ization parameter γ = 0.25, 0.75 fm−1), with the KVP at second order (KVPII), and the results of Refs. [2] and [33]. . . 72
4.10 Same as Tab.4.9 but at E¯nlab= 3 MeV. . . 72
C.1 First eigenvalues (energies are in MeV) obtained solving the bound state problem with β = 1.2 fm−1. The first eigenvalue E0corresponds to the bound state of the two-nucleon system. The Gaussian potential of Eq. (C.1) is used. The results of the calculation of tan δnwith the integral relations, at the energies En, n = 1, 2, 3, are reported in the first line, the second-order estimate is reported in the second line and the exact results of the calculation from Ref. [1] can be found in the third line. The value of the regularization parameter used is γ = 0.8 fm−1. . . 83
F.1 Second-order estimate of the scattering parameters calculated using asymptotic states k = 1, 2 and the average of the two calculations, for Nα = 8 with maximum grand-angular momentum Kαmax = 160
Introduction
The study of the scattering states in few-nucleon systems is of great interest for two reasons: besides delivering the necessary input to study nuclear reactions, it provides also information about the nuclear interaction, since it allows to perform a strong test for the different models.
In this thesis we intend to calculate the scattering parameters, such as phase shifts and mixing angles, using a method based on integral relations. This has been derived first in Refs. [1,2], which represent our starting point. We will repeat the derivation of the integral relations, which starts from the Kohn variational principle (KVP) [3], a principle that can itself be used to calculate scattering states. In fact, we will review the integral relations and the KVP in great detail. Here we want only to mention the advantage of using the integral relations respect to the KVP. In the simplest case of a single scattering parameter, the method based on the integral relations consists in calculating the tangent of the phase shift using a quotient of two integrals [1]. These integrals are of the type hΨ|H − E|ΩF,Gi, where Ψ is the wave function of the system and ΩF,G are the asymptotic solutions of the Schrödinger equation, hence (H − E)ΩF,G → 0 at increasing distance. As a consequence, the integral relations have short range character and the wave function Ψ to be used in the calculation can be any solution of the Schrödinger equation, (H − E)Ψ = 0 in the internal region, where the particles are close and strongly interact with each other. For the functions Ψ, there is no need to specify the asymptotic part. On the contrary, the application of the KVP requires the calculation of matrix elements of the type hΩF,G|H − E|ΩF,Gi. This can be difficult with charged incident particles
and when phenomenological non-local potentials, which present a long-range tail in
r-space, are considered. In fact, the KVP in conjunction with these potentials has
never been used. On the other hand, in practice, within the method of the integral relations, bound-state-like wave functions are used in the calculations. They are the solutions of the Schrödinger equation in the core region and they approach zero at long distance. The calculation of matrix elements hΩF,G|H − E|ΩF,Gi is not
required at first order.
In this thesis we perform a feasibility test of the method based on integral rela-tions (i.e. using bound-state-like wave funcrela-tions to describe the scattering states). We perform this test using the hyperspherical harmonics method (HH) [4,5] to solve the bound state problem, in order to construct the bound-state-like wave functions. The HH method is one of the most powerful ab initio methods used to study the
bound state problem for nuclear systems with A ≤ 4. It has a great versatility, since it can be used in conjunction with two- and three-body, local or non-local potential models. Its extension to a greater number of nucleons is also very promising [6]. Furthermore we should point out that other accurate and well established methods, different from the HH method, exist for A > 4. Our feasibility test should be there-fore of interest also in view of a possible use of the integral relations in conjunction with these methods.
In our study we have first considered the two-nucleon system (A = 2). In this case, we have solved the bound state problem using the HH method with two very different potential models: the Argonne v18(AV18) [7] and the chiral Idaho potential model at next-to-next-to-next-to-leading order (N3LO) [8,9]. We choose these two potentials in order to make contact with the results of the scattering parameters found in the literature [7, 9] with completely different techniques. The AV18, the N3LO as well as the AV14 (see below) potential models will be discussed briefly in the next section.
We have then considered the three-nucleon system (A = 3). In this case we have solved the bound state problem within the HH method using the Argonne v14 (AV14) potential model [10] for the state Jπ = 12+. The results of the scattering parameters calculated using the AV14 potential and with the integral relations can be already found in the literature [2]. However they have been obtained using the pair-correlated hyperspherical harmonics method (PHH) [4] to solve the bound state problem. The PHH method takes into account the correlations of the particles at short distance, due to the hard repulsion of the potential, introducing a correlation factor. As the correlation factor accelerates the convergence, the wave function is constructed with a small number of terms respect to the “uncorrelated” HH method. This could seem an advantage. However the PHH method has never been used in conjunction with non-local potentials, which instead can be treated within the HH approach.
We conclude mentioning how this thesis is organized. In Chapter 2 we review the HH method for the general A-nucleon system [4,5]. We present the bound state wave function, which is constructed as an expansion over a certain number of what are typically called “channels” (i.e. the combinations of quantum numbers that are compatible with the quantum numbers of the state that we are studying) and HH functions. Then the bound state problem is solved variationally, applying the Rayleigh-Ritz variational principle. We discuss also the variational solution of the scattering problem, obtained applying the KVP. We then present two derivations of the integral relations: one is obtained starting from the KVP following Refs. [1,2]. A second one is obtained using directly the bound-state-like wave functions, as explained in Refs. [11, 12, 13]. At this point, we present the implementation of a second-order calculation of the scattering parameters, which however has the drawback of restoring the calculation of the hΩF,G|H − E|ΩF,Gi matrix elements.
In Chapter 3 we apply the whole formalism to the A = 2 case. The bound-state-like wave functions are calculated solving the bound state problem with the AV18 and N3LO potential models, in order to compare our results with those found in the literature [7, 9]. In Chapter 4, we apply the formalism to the A = 3 case concentrating on the neutron-deuteron scattering state Jπ = 12+, under the deuteron break-up threshold. In the three-nucleon case the study is more involved, both in
the formalism and computationally. This is most of all due to the large number of expansion terms that must be included. This is the main reason why we have limited our calculation to the Jπ = 12+ scattering state. In order to make contact with the results found in the literature [2], the scattering parameters are calculated using the AV14 potential. Finally, the conclusions and our outlook for the present work can be found in Chapter5.
1.1
Potential models
The Argonne v18 (AV18) [7] and v14 (AV14) [10] potentials are phenomenological two-nucleon potentials, the first one being the evolution of the second one. The AV18 potential is a high-quality potential model, with a χ2/datum ∼ 1 for all the
large A = 2 database. The AV14 potential is less accurate, since it does not retain charge-symmetry breaking terms (see below) as the AV18. They share though some common features: they contain a short-range distance hard repulsion, an intermediate-range attractive part and a long-range one-pion-exchange term. They are written in an operatorial form, as a sum over 18 (AV18) or 14 (AV14) operators. The AV14 operators are the first 14 ones of the AV18.
Explicitly, the AV14 potential is written in the form
v14,ij = 14 X p=1 vpS(rij) + vIp(rij) + vπp(rij) Oijp , (1.1) where rij is the modulus of the relative distance between the ij particles, rij =
rj − ri, ri and rj being the coordinates of particles i and j, respectively, vpS(rij)
and vIp(rij) are radial components that describe the short- and the intermediate-range part of the potential and they are both treated phenomenologically, that is they are constructed in terms of known functions containing parameters to be fitted with the experimental data. Finally, vpπ(rij) is the well-known radial component that describes the one-pion-exchange part of the nuclear potential at long distance. The operators Oijp of Eq. (1.1) are written explicitly as
Oijp=1,14= [1, σi· σj, Sij, L · S, L2, L2(σi· σj), (L · S)2] ⊗ [1, τi· τj] , (1.2)
where σi,j and τi,j are the spin and isospin Pauli matrices of the particles i and j, respectively, S = σi+σj
2 is the total spin, L is the relative orbital angular momentum and Sij is defined as Sij = 3(σi· ˆrij)(σj· ˆrij) − σi· σj. We notice that these terms
can be constructed starting from the most general interaction that depends on (ri, rj, pi, pj, σi, σj), where pi,j are the momenta of the particles, and imposing
symmetries and invariance properties such as translational invariance, rotational invariance, invariance under Galileo transformations, under particles exchange, as well as parity and time reversal symmetries and the hermitian condition for the potential operator [14]. The term σi· σj can be rewritten in terms of the total spin
S, therefore it takes into account the spin dependence of the potential (different
interaction for singlets and triplets). Sij is a tensor term that takes into account the
fact that the nuclear interaction is not central (L is not a good quantum number and, for example, the deuteron is a np state with quantum number L = 0, 2).
Finally, the spin-orbit term L · S can be observed in the scattering of polarized protons by a spinless target nucleus (the spin and the orbital momentum can be parallel or antiparallel depending on the direction in which the proton travels). The other terms of higher order in Eq. (1.2) are needed to fit the experimental data for scattering waves with L ≥ 3. We notice that in the form of Eq. (1.2) the potential is also invariant under rotations in the isospin space, hence it is isospin independent. Therefore, the AV14 potential is not able to reproduce the (small) differences between the pp, np and nn scattering data. The isospin dependence of the nuclear potential is taken into account in the AV18 potential model. The AV18 potential can be expanded in a similar way to the AV14 potential, as follows
v18,ij = 18
X
p=1
vp(rij)Opij, (1.3)
where the first 14 operators are those in Eq. (1.2), while the other operators Oijp with p = 15, 16, 17, 18 represent the characteristic charge-independence breaking part of the AV18 potential. Specifically,
Op=15,16,17,18ij = [Tij, (σi· σj)Tij, SijTij, τzi+ τzj] , (1.4)
where the isotensor operator Tij is defined as Tij = 3τziτzj− τi· τj. Furthermore,
the AV18 contains a complete electromagnetic potential at the long-range distance. In this way, unlike the AV14, the AV18 potential is able to fit both np and pp data with a χ2/datum ∼ 1.09, for 4301 pp and np data in the energy range up to 350
MeV.
The chiral N3LO Idaho (N3LO) potential model [8,9] is derived in a completely different approach, starting from the more fundamental level of quantum chromo-dynamics (QCD) in the low-energy regime, which is the characteristic regime of nuclear physics. Quarks interact with gluons and the force between the nucleons could be seen as a residual color interaction. However, describing the nuclear forces directly with quarks and gluons is a very complicated problem, because in the low-energy range QCD cannot be studied perturbatively. An efficient approach is given by the formulation of an effective field theory, which in fact is based on a Weinberg seminal idea [15]. The nuclear effective Lagrangian is the most general Lagrangian written consistently with the symmetries of QCD (and implementing explicitly the broken chiral symmetry of QCD), with pions (π) and nucleons (N ) as effective degrees of freedom (in place of the fundamental quarks and gluons). Then the problem is solved perturbatively (the so-called chiral perturbation theory), expand-ing the Lagrangian in powers of ΛQ
χ, where Q is a small external momentum (or pion
mass) and Λχ ≈ 1 GeV represents the chiral-symmetry breaking scale. By writing the effective nuclear Lagrangian as a sum of interaction terms, each term can be expanded in powers (ΛQ
χ)
ν, and the different contributions are derived at different
orders: the leading order (LO) corresponds to the power ν = 0, the next-to-leading order (NLO) to the power ν = 2, because the terms with ν = 1 are excluded by parity and time reversal symmetry. Then the expansion proceeds until the next-to-next-to-next-to-leading order (N3LO) or even more. Some characteristics of the nuclear force emerge even at LO, such as the tensor force. At NLO, as an example, we can find a good description of the intermediate-range interaction even if with
insufficient attraction, which is achieved at the next order. The charge-symmetry breaking terms are included. Finally, the potential at N3LO is able to reproduce the N N scattering database with an accuracy comparable with that of the phe-nomenological AV18 [8]. As a final remark on the chiral potentials, we notice that they are typically derived in momentum-space and they need to be regularized at large momentum. The function used for this leads typically to non-local potential models, as the N3LO model adopted here [8,9].
In this thesis we will mention also another non-local potential model, the CD-Bonn potential [16]. This is a phenomenological potential based on the one-boson-exchange approach, which has been widely used over the years in alternative to the AV18 and is also able to reproduce the N N scattering database with χ2/datum ∼ 1.
The hyperspherical harmonics
method for the A-body system
In this chapter we introduce the hyperspherical harmonics method for a general A-nucleon system. We solve both bound and scattering problem variationally, using two different variational principles. In order to calculate the scattering parameters, we will derive integral relations in two different ways, implementing also a second-order calculation.2.1
Jacobi and hyperspherical coordinates
The Hamiltonian for a system of A interacting nucleons can be written as [5]
H = T + V = A X i=1 p2 i 2mi + V , (2.1)
where T is the non-relativistic kinetic energy operator and V is the potential energy operator. Each nucleon is described by mi, ri and pi, which are the mass, the
spatial coordinate and the momentum of the i-th particle, respectively. The most general interaction potential is a sum of terms that includes contributions from forces between two particles (Vij) or more than two particles (Vijk, . . . )
V = A X i<j Vij + A X i<j<k Vijk+ . . . . (2.2)
In this work we will take into account only two-body interactions, for which our results can be tested with what is available in the literature, and we will neglect the others, which means retaining only the first term of the summation in Eq. (2.2).
The time-independent Schrödinger equation H |ψi = Etot|ψi projected on the coordinate space (r1, . . . , rA) assumes the form
" −~ 2 2 A X i=1 ∇2 ri mi + A X j>i=1 Vij # ψ(r1, . . . , rA) = Etotψ(r1, . . . , rA) , (2.3)
where pi has been replaced by the differential operator −i~∇ri.
In order to simplify the problem we define the total mass M = PA
i=1mi, the
center-of-mass coordinate R = M1 PA
i=1miri and the Jacobi coordinates as [17]
xi= A X j=1 cijyj, i = 1, . . . , N (N = A − 1) , (2.4a) xA= A X j=1 cAjyj. (2.4b)
They are linear combinations of coordinates y1, . . . , yA (yi =√miri, i = 1, . . . , A)
and the last cAjcoefficients are fixed so that xA= R, i.e. cAj=
√
mj
M . The definition
of Eqs. (2.4) leads to the following relation between gradients:
∇y i = A X k=1 cki∇xk. (2.5)
If we impose two more conditions on the coefficients cij, specifically [17]
A X i=1 ckicki = 2 mref k = 1, . . . , N , A X i=1 ckicli = 0 k, l = 1, . . . , A (k 6= l) , (2.6)
where mref is a reference mass, then the kinetic term in Eq. (2.3) can be rewritten in the simple form
T = −~ 2 2 A X i=1 ∇2 ri mi = −~ 2 2 A X i=1 ∇2 yi = − ~2 mref N X i=1 ∇2 xi− ~2 2M∇ 2 R. (2.7)
The complete separation between internal and center-of-mass degrees of freedom is in fact the advantage of using the Jacobi coordinates. The wave function and the total energy in Eq. (2.3) can be rewritten as ψ(r1, . . . , rA) = Ψ(x1, . . . , xN)Φ(R)
and Etot = E + Ecm, respectively, so that, inserting Eq. (2.7) in Eq. (2.3), the Schrödinger equation separates into
" − ~ 2 mref N X i=1 ∇2 xi+ A X j>i=1 Vij # Ψ(x1, . . . , xN) = EΨ(x1, . . . , xN) , (2.8) " − ~ 2 2M∇ 2 R # Φ(R) = EcmΦ(R) . (2.9)
In this way, the internal motion is completely singled out from the center-of-mass motion. Explicit solutions for Eq. (2.9) are (Ω is a normalization volume)
Φ(R) = √1 Ωe i ~P ·R, Ecm = P2 2M . (2.10)
From now on we will work in the center-of-mass reference frame and we will consider only Eq. (2.8).
A solution of Eqs. (2.6) is represented by [4] xN −j+1= s 2mj+1Mj (mj+1+ Mj)mref (rj+1− Rj) , j = 1, . . . , N , (2.11) where Mj = Pj
i=1mi and Rj = M1jPji=1miri are the total mass and the
center-of-mass coordinate of the subsystem of particles 1, . . . , j, respectively. When the masses of all the particles are equal, that is m1 = · · · = mA= m, we can write
xN −j+1 = s 2j j + 1 m mref (rj+1− Rj) , j = 1, . . . , N , (2.12)
where now Mj = jm and Rj = 1jPji=1ri. If the A particles can be clustered
in more than one different way (as for A = 4, for instance), then there can be more than one set of Jacobi coordinates (2.11) [or (2.12)]. This is not the case for
A ≤ 3. Permutations of the A particles must also be taken into account in the
definition (2.11) [or (2.12)], as we will explain later.
The 3N -dimensional space of the internal Jacobi coordinates (x1, . . . , xN) can
be described by the variables (x1, . . . , xN, ˆx1, . . . , ˆxN), where xi is the modulus of
the Jacobi vector xi and ˆxi = (θi, φi) refers to the couple of spherical polar angles
related to the same vector xi, for i = 1, . . . , N . An alternative description is given by the hyperspherical coordinates. In addition to the 2N polar angles ˆxi already
defined, we make use of an hyperradius ρ and N − 1 hyperangles ϕi, whose standard
definition is [4] ρ =qx2 1+ · · · + x2N, ϕi = arctan p x2 1+···+x2i−1 xi , i = 2, . . . , N , (2.13) so that 0 ≤ ϕi ≤ π
2. The inverse transformation can be written explicitly as follows
xN = ρ cos ϕN,
xj = ρ sin ϕN. . . sin ϕj+1cos ϕj,
x1 = ρ sin ϕN. . . sin ϕ2,
(2.14)
with j = 2, . . . , N − 1. The hyperspherical coordinates are written collectively as
(ρ, ΩN) = (ρ, ˆx1, . . . , ˆxN, ϕ2, . . . , ϕN) . (2.15)
The advantage of using the hyperspherical coordinates is evident by looking at the kinetic term in Eq. (2.8), which can be rewritten as [17]
− ~ 2 mref N X i=1 ∇2x i = − ~2 mref N X i=1 " ∂2 ∂x2i + 2 xi ∂ ∂xi −` 2 i(ˆxi) x2i # (2.16a) = − ~ 2 mref " ∂2 ∂ρ2 + 3N − 1 ρ ∂ ∂ρ + Λ2N(ΩN) ρ2 # . (2.16b)
Eq. (2.16a) comes from the usual expression of the Laplace operator ∇2x
i in spherical
coordinates (xi, ˆxi), `2i(ˆxi) being the square of the angular momentum operator
related to the Jacobi vector xi, specifically
− `2 i(ˆxi) = 1 sin2θi " sin θi ∂ ∂θi sin θi ∂ ∂θi ! + ∂ 2 ∂φ2i # . (2.17)
Eq. (2.16b) can be obtained directly from Eq. (2.16a) by expressing the partial derivatives in terms of the hyperspherical coordinates [17]
∂ ∂xi = ∂ρ ∂xi ∂ ∂ρ + D1i, ∂2 ∂x2i = ∂ρ ∂xi 2 ∂2 ∂ρ2 + ∂2ρ ∂x2i ! ∂ ∂ρ + D2i, (2.18)
where D1i and D2iare differential operators that contain terms of derivatives with respect to at least one hyperangle. Substituting Eq. (2.18) in Eq. (2.16a) we obtain
N X i=1 ∇2 xi = ∂2 ∂ρ2 + 3N − 1 ρ ∂ ∂ρ+ N X i=1 D2i+ 2 xi D1i− `2i(ˆxi) x2i ! . (2.19)
Finally, from the above expression in parentheses, we can easily extract the recursive definition of the operator Λ2N(ΩN) that appears in Eq. (2.16b), starting from N = 1,
for which Λ21(Ω1) = −`21(ˆx1), to a generic value of N , so that
Λ2N(ΩN) = ∂ 2 ∂ϕ2N + AN ∂ ∂ϕN −` 2 N(ˆxN) cos2ϕ N +Λ 2 N −1(ΩN −1) sin2ϕN , (2.20)
where AN = [3(N − 2) cot ϕN+ 2(cot ϕN− tan ϕN)]. ΛN(ΩN) is called
grand-angular momentum operator or generalized grand-angular momentum operator.
2.2
Hyperspherical harmonic functions
The wave function in Eq. (2.8) should be rewritten in terms of the hyperspherical variables too. To this purpose, we search for the eigenfunctions of the grand-angular momentum operator of Eq. (2.20) solving the equation [4]
Λ2N(ΩN)Y[K]K(ΩN) = −K(K + 3N − 2)Y[K]K(ΩN) . (2.21)
The functions Y[K]K(ΩN) are the so-called hyperspherical harmonic (HH) functions and are eigenfunctions of the operator Λ2N with eigenvalue −K(K + 3N − 2). K is the grand-angular momentum quantum number and [K] refers to a set of quantum numbers that specifies the function uniquely. In order to derive Eq. (2.21) [17], let us introduce h[K], which is a homogeneous polynomial in the 3N -cartesian coordinates of the N Jacobi vectors xi, where K is the order of the polynomial. By definition of a
separated and we can write h[K] = ρKY[K](ΩN). The polynomial is harmonic, therefore ∆h[K]= 0, where ∆ =PN i=1∇2xi. As a consequence 0 = ∆h[K]= ∆ρKY[K](ΩN) =hK(K + 3N − 2) + Λ2N(ΩN) i ρK−2Y[K](ΩN) . (2.22) This leads to Eq. (2.21).
In the case N = 1 (A = 2), we have
Λ21(Ω1)Y[K]K(Ω1) = −K(K + 1)Y[K]K(Ω1) . (2.23) Since Λ21(Ω1) = −`21(ˆx1), the eigenfunctions Y[K]K(Ω1) are the usual spherical har-monics defined in the 3-dimensional space, K corresponds to the orbital angular momentum quantum number `1, and the functions are completely specified adding the quantum number related to the orbital angular momentum projection on the third axis, [K] = [`1, m1].
In the general case, the eigenfunctions Y[K]K(ΩN) with the definition K ≡ KN,
are calculated in terms of N − 1 functions YKN −1
[KN −1](ΩN −1). They can be explicitly
written as [4]
Y[K]K(ΩN) = F (cos 2ϕN)(cos ϕN)`N(sin ϕN)KN −1Y`NmN(ˆxN)Y
KN −1
[KN −1](ΩN −1) ,
(2.24) where F (cos 2ϕN) is a function to be determined and Y`NmN(ˆxN) is the spherical
harmonic related to the Jacobi vector xN. Inserting the ansatz (2.24) in Eq. (2.21), using the recursive definition (2.20) of the operator Λ2N(ΩN) together with the eigenvalue equations (with the notation ~ = 1)
`2N(ˆxN)Y`NmN(ˆxN) = `N(`N+ 1)Y`NmN(ˆxN) , Λ2N −1(ΩN −1)YKN −1 [KN −1](ΩN −1) = −KN −1(KN −1+ 3N − 5)Y KN −1 [KN −1](ΩN −1) , (2.25)
we obtain the following differential equation for F (cos 2ϕN) ≡ F (z)
(1 − z2)F00+ (α − zβ)F0+ γ = 0 , (2.26) where α = `N − KN −1− 3N 2 + 3 , β = `N + KN −1+ 3N 2 , γ = 1 4[K(K + 3N − 2) − (`N+ KN −1)(`N+ KN −1+ 3N − 2)] . (2.27)
The solution of Eq. (2.26) is proportional to the so-called Jacobi polynomial Pn(a,b)(z)
of order n (n = 0, 1, 2, . . . , −1 ≤ z ≤ 1) [20], with
b − a = α , b + a + 2 = β , n(n + b + a + 1) = γ . (2.28)
Therefore, for K = 2nN + `N + KN −1, the solution of Eq. (2.26) is
F (z) = NνN,`N
nN P
(νN −1,`N+12)
where νN = K +3N2 − 1 and νN −1= KN −1+3(N −1)2 − 1. More details on the Jacobi
polynomials are given in AppendixA.1.
We are now able to rewrite the HH functions from Eq. (2.24) as
Y[K]K(ΩN) = " N Y i=1 Y`imi(ˆxi) #" N Y j=2 jPKj−1,`j nj (ϕj) # , (2.30)
where Y`imi(ˆxi) are the spherical harmonics related to the Jacobi vector xi and jPKj−1,`j nj (ϕj) = N νj,`j nj (cos ϕj) `j(sin ϕ j)Kj−1P (νj−1,`j+12) nj (cos 2ϕj) , (2.31)
with quantum numbers
Kj = j X k=1 (2nk+ `k) , n1= 0 , (2.32) νj = Kj+ 3j 2 − 1 , νj−1= Kj−1+ 3(j − 1) 2 − 1 , (2.33) and the normalization factor Nνj,`j
nj is given by Nνj,`j nj = s 2νjΓ(νj − nj)nj! Γ(νj− nj− `j −1 2)Γ(nj+ `j+ 3 2) . (2.34)
The HH functions are completely specified by the set of 3N − 1 quantum numbers [K] = [`1, m1, . . . , `N, mN, n2, . . . , nN] (2.35)
and the grand-angular quantum number is K = KN.
We briefly discuss two of the main properties of the set of HH functions: or-thonormality and completeness [4]. If the HH functions in Eq. (2.30) are normalized using Eq. (2.34), then the following orthonormality relation holds
Z dΩN
h
Y[K]K(ΩN)i∗Y[KK00](ΩN) = δ[K][K0]δKK0. (2.36) In the above equation dΩN indicates the surface element of the hypersphere of
unit hyperradius and can be extracted from the following expression of the volume element in terms of the hyperspherical coordinates
d3x1. . . d3xN = " N Y j=2 dϕj(sin ϕj)3j−4(cos ϕj)2 #" N Y j=1 dˆxj # dρ ρ3N −1 = dΩNdρ ρ3N −1. (2.37)
Then Eq. (2.36) can be easily obtained taking into account the orthornormality property of the spherical harmonics and the orthogonality property of the Jacobi polynomials (A.2). Completeness relation is written explicitly as
X [K] h Y[K]K(Ω0N)i ∗ Y[K]K(ΩN) = δ3N −1(ΩN − Ω0N) = N Y j=1 δ2(ˆxj− ˆx0j) N Y j=2 δ(ϕj− ϕ0j) (sin ϕj)3j−4(cos ϕj)2 (2.38)
and it also can be demonstrated starting from the expression Z d3q1. . . d3qN ei PN j=1qj·(xj−x 0 j)= (2π)3N N Y j=1 δ3(xj− x0j) , (2.39)
where qj are the conjugate momenta of the Jacobi coordinates xj. Rewriting the equation in terms of the hyperspherical coordinates associated to the Jacobi vec-tors xj and qj, (ρ, ΩN) and (Q, ΩqN) respectively, and using the expansion of the plane wave in the 3N -dimensional space with the hyperspherical coordinates, the completeness relation of Eq. (2.38) follows after some calculations.
The set of HH functions of Eq. (2.30) is a complete and orthonormal set, there-fore it can be used as a basis to expand a regular function of the hyperangular vari-ables ΩN. In particular, it can be used to expand a generic function of the Jacobi
coordinates x1, . . . , xN, such as the A-nucleon system wave function in Eq. (2.8),
with coefficients that depend on the hyperradius
Ψ(x1, . . . , xN) =
X
[K]
a[K](ρ)Y[K]K(ΩN) . (2.40)
2.3
Bound state problem
2.3.1 Bound state wave function
Before being able to write explicitly the A-body wave function that appears in the Schrödinger equation (2.8), we have to discuss two other subjects: angular momentum coupling and symmetry of the wave function.
The most general wave function that describes the A-nucleon system must con-tain not only the spatial but also the spin and isospin degrees of freedom. Total spin and total isospin angular momentum operators are defined as S = s1+ · · · + sA
and T = t1 + · · · + tA, respectively. Total orbital angular momentum operator is
L = `1+· · ·+`N and we define the total angular momentum operator as J = L+S.
The definitions of their projection on the z-axis are straightforward. The total wave function of the A-nucleon system is an eigenfunction of the operators J2 and Jz,
together with the parity operator Π, therefore the expansion (2.40) is not directly applicable and a recouple of the angular momenta is needed.
First we focus on the orbital angular momentum. The definition of the HH functions in Eq. (2.30) in terms of the projection of a state on the hyperangular coordinates is given by
Y[K]K(ΩN) = hΩN|`1, m1, . . . , `N, mN, n2, . . . , nNi . (2.41)
The HH functions contain a product of spherical harmonics, so they are eigenfunc-tions of `2i and `zi with eigenvalues `i(`i+ 1) and mi, respectively (i = 1, . . . , N ).
Following the general rules of angular momentum coupling, we can write them as eigenfunctions of L2 and Lz
which are explicitly given by Y[K]LMK (ΩN) = X m1...mN (`1m1`2m2|L2M2) (L2M2`3m3|L3M3) × . . . (LN −1MN −1`NmN|LM ) Y[K]K(ΩN) , (2.43)
where Mj = Pji=1mi for j = 2, . . . , N − 1 and the sum contains the product of
Clebsh-Gordan coefficients. The new set of relevant quantum numbers is now
[K]LM = [`1, . . . , `N, L2, . . . , LN −1, n2, . . . , nN]LM (2.44)
and it is convenient to put Eq. (2.43) in a more compact form as
Y[K]LMK (ΩN) = hh . . .hhY`1(ˆx1)Y`2(ˆx2) i L2 Y`3(ˆx3) i L3 . . .i LN −1 Y`N(ˆxN) i LM × N Y j=2 jPKj−1,`j nj (ϕj) , (2.45)
where we have used Eq. (2.30). Then the following eigenvalue equations are satisfied [5]
Λ2N(ΩN)Y[K]LMK (ΩN) = K(K + 3N − 2)Y[K]LMK (ΩN) , (2.46)
L2(ΩN)Y[K]LMK (ΩN) = L(L + 1)Y[K]LMK (ΩN) , (2.47)
Lz(ΩN)Y[K]LMK (ΩN) = M Y[K]LMK (ΩN) . (2.48)
The spin and isospin angular momenta behave similarly. We define the eigen-functions of the spin (isospin) operators s2i and siz (t2i and tiz) as
χsimsi(i) = hσi|simsii ηtimti(i) = hτi|timtii , i = 1, . . . , A , (2.49)
which satisfy
s2iχsimsi(i) = si(si+ 1)χsimsi(i) sziχsimsi(i) = msiχsimsi(i) , (2.50)
t2iηtimti(i) = ti(ti+ 1)ηtimti(i) tziχtimti(i) = mtiχtimti(i) , (2.51)
with si = 12 and msi = ± 1 2 (ti = 1 2 and mti = ± 1
2), because nucleons are fermions with half-integer spin12 and they belong to the representation of isospin of dimension 2 (doublet). Eigenfunctions of total spin and isospin angular momentum can be cast in the form χ[S]SMS =hh. . .hhχ1 2(1)χ 1 2(2) i S2 χ1 2(3) i S3 . . .i SN χ1 2(A) i SMS , (2.52) η[T ]T MT = hh . . .hhη1 2 (1)η1 2 (2)i T2 η1 2 (3)i T3 . . .i TN η1 2 (A)i T MT , (2.53)
with [S]SMS = [S2, . . . , SN]SMS and [T ]T MT = [T2, . . . , TN]T MT. The functions
defined in Eqs. (2.52) and (2.53) satisfy the eigenvalue equations
S2χ[S]SMS = S(S + 1) χ[S]SMS Szχ[S]SMS = MSχ[S]SMS, (2.54)
At last, recoupling Eqs. (2.45) and (2.52) and adding Eq. (2.53), we define YJ Jzπ [µ]KLST(ΩN) = Y[K]LK (ΩN)χ[S]S J Jzη[T ]T MT, [µ] = [K][S][T ] , (2.56)
which can be used to expand the A-nucleon wave function. To be noticed that the following eigenvalue equations hold [5]
J2YJ Jzπ [µ]KLST(ΩN) = J (J + 1)Y J Jzπ [µ]KLST(ΩN) , (2.57) JzYJ J[µ]KLSTzπ (ΩN) = JzYJ J[µ]KLSTzπ (ΩN) , (2.58) ΠYJ Jzπ [µ]KLST(ΩN) = (−1) KYJ Jzπ [µ]KLST(ΩN) . (2.59) The last equation follows from the fact that under parity operation, among all the hyperpherical variables, only the 2N polar angles are affected ˆxi = (θi, φi) → (π −
θi, π + φi) (hyperangles and hyperradius are unchanged), therefore every spherical
harmonics in Eq. (2.43) [or Eq. (2.30)] acquires a factor (−1)`i and the overall factor
is
π = (−1)`1+···+`N = (−1)K−(2n2+···+2nN)= (−1)K. (2.60)
The wave function of the A-nucleon system can be finally cast in the form
Ψ(x1, . . . , xN) = X KLST X [µ] f[µ]KLST(ρ)YJ J[µ]KLSTzπ (ΩN) , (2.61)
where the function f[µ]KLST(ρ) contains all the dependence on the hyperradius and the sum runs over all the possible quantum numbers that are compatible with
KLST and a given state of fixed total angular momenta J , Jz and parity π.
As already mentioned, we have to take into account also the symmetry of the wave function. Nucleons are fermions and, according to the Pauli principle, the
A-nucleon system wave function must be antisymmetric under the exchange of any
two particles. This is achieved, first, by writing the wave function as a sum of so-called Faddeev-like amplitudes [4]
ΨA=
Np
X
p=1
Ψ(x(p)1 , . . . , x(p)N ) , (2.62) where the sum runs over the Np even permutations of the A particles. We then observe that the Jacobi vectors change under a permutation of the particles. This can be seen from the definition (2.12) where, for example, for j = 1 we obtain
xN ∝ (r2− r1). However using a different permutation of the particles, xN could
be proportional, for A = 3 as an example, to (r1− r3) or (r3− r2). Therefore, xN is in fact x(p)N ∝ (rj − ri). Turning to hyperspherical coordinates, permutations do
not affect the hyperradius but hyperangular variables change and all the functions that depend on them change consequently. Using the expansion (2.61) we write
ΨA= Np X p=1 X KLST X [µ] f[µ]KLST(ρ)YJ J[µ]KLSTzπ (Ω (p) N ) (2.63)
where now Ω(p)N = (ˆx(p)1 , . . . , ˆx(p)N , ϕ(p)2 , . . . , ϕ(p)N ) and the hyperangular functions (2.56) are redefined as YJ Jzπ [µ]KLST(Ω (p) N ) = Y[K]LK (Ω(p)N )χ(p)[S]S J Jzη (p) [T ]T MT, [µ] = [K][S][T ] . (2.64) Requesting YJ Jzπ [µ]KLST(Ω (p) N ) = −Y J Jzπ [µ]KLST(Ω (pi↔j) N ) , (2.65)
where p is a generic permutation (ij . . . ) and pi↔j corresponds to (ji . . . ), where
particles i and j have been exchanged, the total antisymmetry of the wave func-tion of the system ΨA is guaranteed. This condition can be translated into further condition for the quantum numbers, considering the effect of this particular ex-change of particles on the orbital, spin and isospin functions of Eqs. (2.45), (2.52) and (2.53), respectively. The explicit overall factor of the exchange i ↔ j is (−1)`N(−1)S2+1(−1)T2+1 and the condition (−1) = (−1)`N+S2+T2 implies `
N+ S2+
T2 = odd.
The most important and useful fact is that any HH function (2.45) in a given permutation p can be rewritten as a linear combination of HH functions in a “ref-erence” permutation ¯p with appropriate coefficients. We write [4,21]
YK [K]LM(Ω (p) N ) = X [K0] RKL(p→ ¯[K][K0]p)Y[KK0]LM(Ω( ¯Np)) , (2.66)
where the sum runs over a finite number of combinations of quantum numbers [K0] = [`01, . . . , `0N, L02, . . . , L0N −1, n02, . . . , n0N] with the constraints K0= K and L0 =
L and the coefficients are called transformation coefficients (TC). They are defined
with the help of the orthonormality relation of the HH functions (2.36) on the same permutation [4] RKL(p→ ¯[K][K0]p) = Z dΩ( ¯Np)hY[KK0]LM(Ω( ¯ p) N ) i∗ Y[K]LMK (Ω(p)N ) . (2.67) The actual calculation of the TC can be done in other different ways such as making use of recurrence relation [22].
For the spin and isospin part of the wave function the procedure is similar. We define the spin (isospin) permutated functions in terms of functions given in the reference permutation ¯p [21] χ(p)[S]SM S = X [S0] S[S][SS(p→ ¯0]p)χ ( ¯p) [S0]SM S, η (p) [T ]T MT = X [T0] T[T ][TT (p→ ¯0]p)η ( ¯p) [T0]T M T , (2.68)
where again the sum is over a finite number of combinations of quantum num-bers [S0] = [S20, . . . , SN0 ] with S0 = S ([T0] = [T20, . . . , TN0 ] with T0 = T ) and the transformation coefficients are
S[S][SS(p→ ¯0]p)= hχ ( ¯p) [S0]SM S|χ (p) [S]SMSi , T T (p→ ¯p) [T ][T0] = hχ ( ¯p) [T0]T M T|χ (p) [T ]T MTi . (2.69)
Transformation coefficients of Eqs. (2.66) and (2.68) do not depend on the quan-tum numbers M , MS (and MT) so that we are able to rewrite Eq. (2.64) in terms of the reference permutation ¯p
YJ Jzπ [µ]KLST(Ω (p) N ) = X [µ0] aKLST (p→ ¯[µ][µ0] p)YJ J[µ0z]KLSTπ (Ω ( ¯p) N ) , (2.70)
where the coefficient is a product of TC and [µ] = [K][S][T ], [µ0] = [K0][S0][T0] such that K0 = K, L0 = L, S0 = S, T0 = T . Summing over the permutations we obtain
Np X p=1 YJ Jzπ [µ]KLST(Ω (p) N ) = X [µ0] AKLST [µ][µ0]YJ J[µ0z]KLSTπ (Ω ( ¯p) N ) , (2.71) with coefficients AKLST [µ][µ0] = Np X p=1 aKLST (p→ ¯[µ][µ0] p). (2.72) The total wave function of Eq. (2.63) becomes
ΨA= X KLST X [µ] f[µ]KLST(ρ)X [µ0] AKLST [µ][µ0]YJ J[µ0z]KLSTπ (Ω ( ¯p) N ) . (2.73)
In order to solve the bound state problem, the hyperradial functions f[µ]KLST(ρ) are in turn expanded on a basis of orthonormal functions as follows
f[µ]KLST(ρ) =
NL
X
m=1
cm[µ]KLSTfm(ρ) . (2.74)
Here the sum is truncated for computational reasons and in this work we use for
fm(ρ) the functions fm(ρ) = β 3N 2 s m! (m + 3N − 1)!L (3N −1) m (βρ) e −βρ 2 , (2.75)
where L(3N −1)m (βρ) are the generalized Laguerre polynomials defined in Appendix
A.2. The normalization factors in Eq. (2.75) guarantee that
Z ∞
0
dρ ρ3N −1fm(ρ)fm0(ρ) = δmm0, (2.76) which is a consequence of the orthogonality relation of the Laguerre polynomials (A.5) (with x = βρ and α = 3N − 1) and β is a non linear parameter. With the exponential factor, these functions have also the property fm(ρ) → 0 for hyperradius
ρ → ∞.
At the end, inserting the expansion (2.74), with definition of Eq. (2.75), in Eq. (2.73) we obtain ΨA= X KLST X [µ] NL X m=1 cm[µ]KLSTfm(ρ) X [µ0] AKLST [µ][µ0] YJ J[µ0z]KLSTπ (Ω ( ¯p) N ) . (2.77)
Sometimes another arrangement of the set of quantum numbers [µ]KLST is useful. For this reason we define
[α][nα] = [`1α, . . . , `N α, [Lα], Lα, [Sα], Sα, [Tα], Tα][n2α, . . . , nN α] , (2.78)
where [Lα] ≡ [L2α. . . L(N −1)α], [Sα] ≡ [S2α. . . SN α], [Tα] ≡ [T2α. . . TN α] and the
subscript α refers to what we call a “channel”. In this way the total wave function of Eq. (2.63) can be rewritten in the equivalent compact form
ΨA= Np X p=1 Nα X α=1 X [nα] fα[nα](ρ)YJ Jzπ α[nα]Kα(Ω (p) N ) . (2.79)
With Nαwe indicate the number of channels to be included in the calculation. When
a channel is fixed, the hyperangular quantum numbers [nα] vary from a minimum
n0
α to a maximum value nmaxα . The first can be different from zero and this is due
to the fact that, summing over the permutations, some states can appear that are linearly dependent from others that have already been included, so they do not need to be retained. The latter is conveniently chosen for every specific problem that needs to be solved. As a consequence, the grand-angular momentum quantum number Kα varies from Kα0 to Kαmax too. Expanding the hyperradial functions in
Eq. (2.79) over a Laguerre basis, as in Eq. (2.74), we have
ΨA= Nα X α=1 X [nα] NL X m=1 cmα[nα]fm(ρ) Np X p=1 YJ Jzπ α[nα]Kα(Ω (p) N ) , (2.80)
where, obviously, the expression of the sum over permutations of the HH basis in terms of the transformation coefficients (2.71) is still valid (adding subscripts [µα], [µα0] and KαLαSαTα).
Alternatively, in an even more compact form, the A-nucleon wave function with the correct coupling of momenta and with the correct exchange symmetry can be written as
ΨA=
X
mκ
cmκΨmκ, (2.81)
where the index κ refers collectively to all the HH indices, κ = α[nα], and the functions Ψmκ are defined as
Ψmκ= fm(ρ) Np X p=1 YJ Jzπ κ (Ω (p) N ) . (2.82)
2.3.2 Rayleigh-Ritz variational principle
In order to solve the bound state problem, we need to find the solutions of the Schrödinger equation for an A-nucleon system, i.e.
H |ΨAi = (T + V ) |ΨAi = E |ΨAi , (2.83)
with the kinetic energy T as in Eq. (2.16b), the potential V and the wave function ΨAas in Eq. (2.81). Instead of solving Eq. (2.83), we solve the bound state problem
using a variational procedure, namely using the Rayleigh-Ritz principle. We review the Rayleigh-Ritz variational principle in AppendixB.1. The truncated expansion of the hyperradial functions over a complete orthonormal set of functions, as in Eq. (2.74), allows us to use the functions ΨAas trial wave functions, whose expansion coefficients are determined solving Eq. (B.3), i.e.
hδΨA|H − E|ΨAi = 0 . (2.84)
Explicitly, inserting Eq. (2.81) in the functional hΨA|H − E|ΨAi = 0, we can
per-form the variation about the set of real coefficients {cmκ}, and we obtain
X
m0κ0
hΨmκ| T + V − E | Ψm0κ0i cm0κ0 = 0 . (2.85) The norm and the kinetic energy matrix elements are defined as
hΨmκ|Ψm0κ0i = Nmκ,m0κ0 = δmm0δκκ0, (2.86) hΨmκ|T |Ψm0κ0i = Tmκ,m0κ0 = Tmm0δκκ0, (2.87) where δmm0 on the Laguerre basis indices comes from the orthonormality relation (2.76) and δκκ0 is calculated using Eq. (2.77) with the set of indices [µ]KLST ≡ κ ([µ] = [K][S][T ]) so that we can use the orthonormality property of the functions YJ Jzπ
[µ]KLST(Ω (p)
N ) in the same permutation p = ¯p. Furthermore, Tmm0 is calculated using the expression of the kinetic energy given in Eq. (2.16b) together with the eigenvalue equation (2.21). Finally we have
δκκ0 = Z dΩ( ¯Np) X [σ] AKLST [µ][σ] Y J Jzπ [σ]KLST(Ω ( ¯p) N ) ∗ X [σ0] AK0L0S0T0 [µ0][σ0] Y[σJ J0z]Kπ0L0S0T0(Ω ( ¯p) N ) (2.88) =X [σ] AKLST [µ][σ] ∗ AKLST [µ0][σ] , (2.89) Tmm0 = − ~ 2 mref Z dρ ρ3N −1fm(ρ) ∂2 ∂ρ2 + 3N − 1 ρ ∂ ∂ρ− K(K + 3N − 2) ρ2 fm0(ρ) , (2.90)
with [σ] = [K00][S00][T00] such that K00= K, L00= L, S00= S and T00= T .
The calculation of the potential matrix elements is more involved. First of all we notice that hΨmκ|V |Ψm0κ0i = A X j>i=1 hΨmκ|Vij|Ψm0κ0i = A(A − 1) 2 hΨmκ|V12|Ψm0κ0i , (2.91) where V has been replaced with the first term of the expression of the potential of Eq. (2.2), that is a two-body potential. V12 is the two-body potential acting on a representative couple of particles among all the possible couples ij that can be formed in the A-nucleon system (with j > i). This explains the factor A(A−1)2 .
Potential matrix elements of this type are computed using a different basis which reflects the more convenient jj-coupling of the angular momenta. With the same
definition of quantum numbers that we have already given, instead of using HH functions coupled as in Eq. (2.56) [see also Eqs. (2.45) and (2.52)] we write
ΥJ Jzπ [ν]KT(ΩN) = " . . . h Y`N(ˆxN)χS2 jN Y`N −1(ˆxN −1)χs3(3) jN −1 i JN −1 × Y`N −2(ˆxN −2)χs4(4) jN −2 JN −2 . . . J2 Y`1(ˆx1)χsA(A) j1 # J Jz × η[T ]T MT N Y j=2 jPKj−1,`j nj (ϕj) , (2.92) where χS2 ≡ χS2MS2 = [χs1(1)χs2(2)]S2MS2 (si = 1
2, i = 1, . . . , A), the isospin function η[T ]T MT is given in Eq. (2.53) and jPKj−1,`j
nj (ϕj) is defined in Eq. (2.31).
The new set of quantum numbers is now [ν]KT , with
[ν] = [K]S2[T ][J ] = [`1, . . . , `N, n2, . . . , nN]S2[T2, . . . , TN][j1, . . . , jN, J2, . . . , JN −1] .
(2.93) The basis YJ Jzπ
[µ]KLST is a linear combination of basis Υ
J Jzπ
[ν]KT with coefficients that can be written with the help of the 9j Wigner coefficients. In this way, considering different permutations, summing over them and defining a reference permutation (¯p), following the procedure of Sec.2.3.1, we can write
Np X p=1 YJ Jzπ [µ]KLST(Ω (p) N ) = X [ν] BKLST [µ][ν] Υ J Jzπ [ν]KT(Ω ( ¯p) N ) , (2.94)
where, as a consequence, the coefficients BKLST[µ][ν] are connected to the coefficients AKLST
[µ][µ0] of Eq. (2.72) through the 9j coefficients. Inserting Eq. (2.94) into Eq. (2.82), the functions Ψmκ can be rewritten as
Ψmκ= fm(ρ) X [ν] BKLST [µ][ν] ΥJ J[ν]KTzπ (Ω ( ¯p) N ) = X [ν] BKLST [µ][ν] Φmλ, λ ≡ [ν]KT , (2.95)
which can be inserted in Eq. (2.91) giving
hΨmκ|V12|Ψm0κ0i = X [ν] BKLST [µ][ν] ∗ X [ν0] BK0L0S0T0 [µ0][ν0] hΦmλ|V12|Φm0λ0i , (2.96) with hΦmλ|V12|Φm0λ0i = Z d3x( ¯1p). . . d3xN( ¯p)fm(ρ)fm0(ρ) ×hΥJ Jzπ [ν]KT(Ω ( ¯p) N ) i∗ V (x( ¯Np))ΥJ Jzπ [ν0]K0T0(Ω ( ¯p) N ) . (2.97)
Notice that we have chosen the reference permutation (¯p) of the Jacobi coordinate
x( ¯Np) proportional to the relative distance between particle 1 and 2, x( ¯Np) ∝ r2 −
r1. Integrating out the hyperspherical variables related to the first N − 1 Jacobi coordinates, we obtain delta factors on the quantum numbers
